Multidimensional hyperspace

The concept of property space, which was coined to quanhtahvely describe the phenomena in social sciences [11, 12], has found many appUcahons in computational chemistry to characterize chemical space, i.e. the range in structure and properhes covered by a large collechon of different compounds [13]. The usual methods to approach a quantitahve descriphon of chemical space is first to calculate a number of molecular descriptors for each compound and then to use multivariate analyses such as principal component analysis (PCA) to build a multidimensional hyperspace where each compound is characterized by a single set of coordinates. 

A univariate observation, such as a single laboratory result, may be represented graphically as a point on a line, the axis. The results obtained by two different laboratory tests performed on the same specunen (a bivariate observation) may be displayed as a point in a plane defined by two perpendicular axes. With three results, we have a trivariate observation and a point in a space defined by three perpendicular axes, and so on. We lose the possibility for visualization of a multivariate observation when there are more than three dimensions. StiU, we can consider the multivariate observation as a point in a multidimensional hyperspace with as many mutually perpendicular axes as there are results of different tests. The prefix hyper signifies, in this context, more than three dimensions. Such multivariate observations are also... 

The multidimensional counterpart to Newton s method is Newton-Raphson iteration. A mathematics professor once complained to me, with apparent sincerity, that he could visualize surfaces in no more than twelve dimensions. My perspective on hyperspace is less incisive, as perhaps is the reader s, so we will consider first a system of two nonlinear equations / = a and g = b with unknowns, v and y. 

To a frog with its simple eye, the world is a dim array of grays and blacks. Are we like frogs in our limited sensorium, apprehending just part of the universe we inhabit Are we as a species now awakening to the reality of multidimensional worlds in which matter undergoes subtle reorganizations in some sort of hyperspace ... 

The PSO mimics the said behavior. Every individual of the swarm is considered as a particle in a multidimensional space that has a position and a velocity. These particles fly through hyperspace and remember the best position that they lave seen. Members of a swarm communicate good positions to each other and adjust their own position and velocity based on these good positions. There are two main ways this communication is done swarm best that is known to all and local bests are known in neighborhoods of particles. Updating the position and velocity is done in each iteration as follows ... 

Interlude 2.1 The Abstract Concept of Hyperspace We have introduced hyperspace as a QN multidimensional space. The independent coordinates are the ZN position variables and the ZN momentum variables for the N total molecules in the system. It is impossible to draw such a system in three dimensions, so we must think of hyperspace in abstract or mathematical terms. 

In a second approach, the spectral data are expressed in terms of a vector - for example, using Hadamard or Fourier transform coefficients of IR spectra - each element of which is treated as a coordinate in multidimensional space. Each spectrum occupies a point in hyperspace and the similarity between an unknown and a reference entry is measured by the distance between the two points. Once again, the output is a rank-ordered list of structures corresponding to the spectra producing the smallest distance to the query. 

A second distinction among the different classification methods proposed in the literature concentrates on the mathematical form of the functional relationship representing the classification rules in terms of the measured variables (or, alternatively, on the geometrical shape of the decision boundaries in the multidimensional space). In this framework, the main differentiation is made between linear and non-linear methods, even if the latter can be sometimes further subdivided according to the kind of non-linearity they implement (e.g. quadratic, polynomial, etc.). In linear methods, the classification mles result in decision boundaries, which are linear functions of the original variables (i.e. which correspond to linear surfaces in the hyperspace spanned by the variables a line in two dimensions, a plane in three... 

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